Square root

In mathematics, a square root of a number x is another number that, when multiplied by itself (squared), becomes x. When it is non-negative, it is represented by the symbol ${\displaystyle {\sqrt {x}}}$, and called the principal square root of x.^[1][2] For example, 3 is the square root of 9, because 3×3=9. Only numbers bigger than or equal to zero have real square roots. The only square root of zero is zero.

An important number is the square root of 2, which is an irrational number. Its value is around 1.41421.^[1] It is the length of a diagonal in the square whose side is 1.^[3]

Square roots of negative numbers are not real numbers – they are imaginary numbers. Imaginary numbers are basically numbers whose square is negative. Every complex number, except 0, has 2 square roots. For example, −1 has two square roots. We call them ${\displaystyle i}$ and ${\displaystyle -i}$. We also consider the number ${\displaystyle i}$ as the principal square root of −1, and called it the imaginary unit.

The sign for a square root is made by putting a bent line over a number, like this: ${\displaystyle {\sqrt {4}}}$. This is read as "the square root of 4" (or whatever number you are taking the square root of).

A whole number with a square root that is also a whole number is called a perfect square. The first few perfect squares are: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225.

Below is a table of square roots (rounded to 3 decimal places).

1. ↑ ^{1.0 1.1} "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
2. ↑ Weisstein, Eric W. "Square Root". mathworld.wolfram.com. Retrieved 2020-08-28.
3. ↑ "Irrationality of the square root of 2". www.math.utah.edu. Archived from the original on 2023-06-05. Retrieved 2020-08-28.